Derivatives of Vector Valued Functions

Skip to main content

Section 4.2 Derivatives of Vector Valued Functions

Suppose that

g

is a function with domain

D \subset R^{N}

and values in

R^{k} .

We denote its component functions by

g_{i},

i = 1, \dots, k

and define

\frac{\partial}{\partial x_{j}} g (x) = \frac{\partial}{\partial x_{j}} [\begin{matrix} g_{1} (x) \\ ⋮ \\ g_{k} (x) \end{matrix}] := [\begin{matrix} \frac{\partial}{\partial x_{j}} g_{1} (x) \\ ⋮ \\ \frac{\partial}{\partial x_{j}} g_{k} (x) \end{matrix}] .

We call such a function a vector valued function.

If

N = 1,

and the variable is

t \in R

we set

\frac{d}{d t} g (t) = g^{'} (t) = \frac{d}{d t} [\begin{matrix} g_{1} (t) \\ ⋮ \\ g_{k} (t) \end{matrix}] := [\begin{matrix} g_{1}^{'} (t) \\ ⋮ \\ g_{k}^{'} (t) \end{matrix}],

that is, we differentiate component-wise. We now want to derive an interpretation of

g^{'} (t) .

To do so it is useful to think of

g (t)

to be the position of moving particle at time

t .

Then the difference coefficient

\frac{1}{Δ t} (g (t + Δ t) - g (t))

points in the direction of the line

L

shown in Figure 4.12. The magnitude of that vector is the approximate speed of the particle moving from

g (t)

to

g (t + Δ t) .

Figure 4.12. Secants approaching the tangent of a curve.
🔗

Passing to the limit we see that

\frac{1}{Δ t} (g (t + Δ t) - g (t)) = [\begin{matrix} \frac{g_{1} (t + Δ t) - g_{1} (t)}{Δ t} \\ ⋮ \\ \frac{g_{k} (t + Δ t) - g_{k} (t)}{Δ t} \end{matrix}] \overset{Δ t \to 0}{\to} [\begin{matrix} g_{1}^{'} (t) \\ ⋮ \\ g_{k}^{'} (t) \end{matrix}] .

Let us summarise what we found in the following proposition.

Proposition 4.13. Derivative of vector valued function of a single variable.

Under the above assumption

\begin{matrix} (4.1) & the vector g^{'} (t) is tangential to the path described by g (t) . \end{matrix}

Moreover,

\begin{matrix} (4.2) & lim_{Δ t \to 0} \frac{‖ g (t + Δ t) - g (t) ‖}{Δ t} = ‖ g^{'} (t) ‖ . \end{matrix}

Remark 4.14. Speed and velocity of a particle.

If we consider

g (t)

as the position of a particle at time

t,

then

\frac{‖ g (t + Δ t) - g (t) ‖}{Δ t}

is the approximate speed of that particle travelling from

g (t)

to

g (t + Δ t) .

Now (4.2) tells us that

‖ g^{'} (t) ‖

is very close to that speed. The direction in which the particle travels is the tangential direction given by

g^{'} (t) .

In other words,

g^{'} (t)

is the velocity of the particle at

g (t) .

The distance travelled in a small time interval is therefore

‖ g (t + Δ t) - g (t) ‖ ≊ ‖ g^{'} (t) ‖ Δ t .

Note that the above approximate equality holds no matter what

k

is.

We can also use the above to find a representation of the tangent line to a curve in space.

Proposition 4.15. Equation of tangent to curve.

Suppose that

I \subset R

is an interval, and

g : I \to R^{N}

is a differentiable function. Then

r (s) := g (t) + s g^{'} (t), s \in R

is an equation for the tangent line to the curve described by

g (t)

{\upshape (}

t \in I

{\upshape )} at

g (t) .

For vectors we have three different kinds of products: the multiplication by scalars and the scalar product in

R^{N}

for general

N,

and the cross product in

R^{3} .

We conclude this section by proving a ‘product rule’ for these products.

Theorem 4.16. Generalised product rule.

Suppose that

f

and

g

are vector valued continuous functions defined on an open set

D \subset R^{N}

with values in

R^{k},

and that

u

is a continuous function on

D

with values in

R .

Then

$\frac{\partial}{\partial x_{i}} (u (x) f (x)) = (\frac{\partial u}{\partial x_{i}} (x)) f (x) + u (x) \frac{\partial f}{\partial x_{i}} (x)$
$\frac{\partial}{\partial x_{i}} (f (x) \cdot g (x)) = (\frac{\partial f}{\partial x_{i}} (x)) \cdot g (x) + f (x) \cdot \frac{\partial g}{\partial x_{i}} (x)$
If $k = 3$ then

$\frac{\partial}{\partial x_{i}} (f (x) \times g (x)) = (\frac{\partial f}{\partial x_{i}} (x)) \times g (x) + f (x) \times \frac{\partial g}{\partial x_{i}} (x)$

whenever all relevant derivatives exist.

Proof.

The proof of all the above formulae are analogous to the classical product rule. As an example we give a proof of the first. By definition of the partial derivative we need to compute the limit of the difference quotient

D (t) := \frac{1}{t} (u (x + t e_{i}) f (x + t e_{i}) - u (x) f (x))

as

t \to 0,

where

e_{i}

is the vector

e_{i} := (0, \dots, 0, 1, 0, \dots, 0),

where the

1

appears in the

i

-th coordinate

By adding and subtracting a term we can rewrite

D (t)

by

D (t) = \frac{1}{t} (u (x + t e_{i}) - u (x)) f (x + t e_{i}) + u (x) \frac{1}{t} (f (x + t e_{i}) - f (x)) .

Using the continuity of the functions and the definition of the partial derivatives we get from the above that

lim_{t \to 0} D (t) = (\frac{\partial u}{\partial x_{i}} (x)) f (x) + u (x) \frac{\partial f}{\partial x_{i}} (x)

as required.

web analytics